Spatially misaligned data, where the data are measured at different spatial scales, are a common issue in spatial modelling [1]}. In ecological and epidemiological research for example, it is of interest to estimate the effect of pollution exposures on health outcomes such as incidence of certain diseases [2]}, [3]}, [4]}. Measurements of exposures are typically collected from a network of monitoring stations, from satellites, or from outputs of computer simulations using numerical models describing the origins and dispersion of exposures. Data from monitoring stations are typically sparsely located in the spatial domain, while the two latter measurements are high resolution data on fine regular grids. The former are classified as point-referenced or geostatistical data, while the latter are considered as high-resolution areal data on fine regular grids [5]}. Moreover, data on health outcomes such as the incidence of certain diseases are available as aggregated counts on fixed and irregular areal units. Such data are organized as counts of mortality, morbidity, or hospital admissions by type of disease over a set of regions partitioning the entire study region for each time point [5]}. Although these data derive from individual-level information, which would enable the analysis of spatial pattern of case event locations and to quantify the effects of long-term exposure, individual-level data are usually not available and are also costly to obtain in practice, and hence have been summarised across administrative units [5]}, [8]}, [9]}. As a result, time series and areal study designs have been frequently used to model health outcome data. However, this comes at a cost since the summary statistics on the population across areas cannot be used to assess effects at the individual level. Moreover, there is the danger of assuming that the associations observed from the areal level also hold at the individual level, often referred to as the ecological fallacy [5]}. Linking pollution exposures and health outcomes therefore necessitates inference on a spatial scale that is different from the scale of the original data, referred to as a change of support problem (COSP).
